vi PREFACE

People with mathematical training are entitled to ask for a deeper and more

quantitative understanding of what is going on here. They may feel optimistic about

attaining it from their experience with the older areas of fundamental physics that

have proved very congenial to mathematical study: the differential equations of

classical mechanics, the geometry of Hamiltonian mechanics, and the functional

analysis of quantum mechanics. But when they attempt to learn quantum field

theory, they are likely to feel that they have run up against a solid wall. There are

several reasons for this.

In the first place, quantum field theory is hard. A mathematician is no more

likely to be able to pick up a text on quantum fields such as Peskin and Schroeder

[89] and understand its contents on a first reading than a physicist hoping to do the

same with, say, Hartshorne’s Algebraic Geometry. At the deep conceptual level, the

absence of firm mathematical foundations gives a warning that some struggle is to

be expected. Moreover, quantum field theory draws on ideas and techniques from

many different areas of physics and mathematics. (Despite the fact that subatomic

particles behave in ways that seem completely bizarre from the human perspective,

our understanding of that behavior is built to a remarkable extent on classical

physics!) At the more pedestrian level, the fact that the universe seems to be

made out of vectors and spinors rather than scalars means that even the simplest

calculations tend to involve a certain amount of algebraic messiness that increases

the effort needed to understand the essential points. And at the mosquito-bite level

of annoyance, there are numerous factors of −1, i, and 2π that are easy to misplace,

as well as numerous disagreements among different authors as to how to arrange

various normalization constants.

But there is another diﬃculty of a more cultural and linguistic nature: physics

texts are usually written by physicists for physicists. They speak a different dialect,

use different notation, emphasize different points, and worry about different things

than mathematicians do, and this makes their books hard for mathematicians to

read. (Physicists have exactly the same complaint about mathematics books!) In

the mathematically better established areas of physics, there are books written from

a more mathematical perspective that help to solve this problem, but the lack of

a completely rigorous theory has largely prevented such books from being written

about quantum field theory.

There have been some attempts at cross-cultural communication. Mathemati-

cal interest in theoretical physics was rekindled in the 1980s, after a period in which

the long marriage of the two subjects seemed to be disintegrating, when ideas from

gauge field theory turned out to have striking applications in differential geometry.

But the gauge fields of interest to the geometers are not quantum fields at all, but

rather their “classical” (unquantized) analogues, so the mathematicians were not

forced to come to grips with quantum issues. More recently, motivated by the de-

velopment of string theory, in 1996–97 a special year in quantum field theory at the

Institute for Advanced Study brought together a group of eminent mathematicians

and physicists to learn from each other, and it resulted in the two-volume collection

of expository essays Quantum Fields and Strings [21]. These books contain a lot

of interesting material, but as an introduction to quantum fields for ordinary mor-

tals they leave a lot to be desired. One drawback is that the multiple authorships

do not lead to a consistent and cohesively structured development of the subject.

Another is that the physics is mostly on a rather formal and abstract level; the